Sum of Paths

题意翻译

一条线段上有 n 个点，机器人每次只能移动到自己相邻的点上，且不能离开这 n 个点。每个点都有分数，分别是 $a_1,a_2,\dots,a_n$。 机器人将从任意一个点开始时，在线段上移动 k 次，每到一个点（包括起始点和重复路过的点），机器人的总分就会加上该点的分数。 现在，我们将对这条线段作 q 次单点点权修改，试求出所有的移动方案的机器人总分的总和。

题目描述

There are $ n $ cells, numbered $ 1,2,\dots, n $ from left to right. You have to place a robot at any cell initially. The robot must make exactly $ k $ moves. In one move, the robot must move one cell to the left or right, provided that it doesn't move out of bounds. In other words, if the robot was in the cell $ i $ , it must move to either the cell $ i-1 $ or the cell $ i+1 $ , as long as it lies between $ 1 $ and $ n $ (endpoints inclusive). The cells, in the order they are visited (including the cell the robot is placed), together make a good path. Each cell $ i $ has a value $ a_i $ associated with it. Let $ c_0, c_1, \dots, c_k $ be the sequence of cells in a good path in the order they are visited ( $ c_0 $ is the cell robot is initially placed, $ c_1 $ is the cell where the robot is after its first move, and so on; more formally, $ c_i $ is the cell that the robot is at after $ i $ moves). Then the value of the path is calculated as $ a_{c_0} + a_{c_1} + \dots + a_{c_k} $ . Your task is to calculate the sum of values over all possible good paths. Since this number can be very large, output it modulo $ 10^9 + 7 $ . Two good paths are considered different if the starting cell differs or there exists an integer $ i \in [1, k] $ such that the current cell of the robot after exactly $ i $ moves is different in those paths. You must process $ q $ updates to $ a $ and print the updated sum each time. Each update changes the value of exactly one cell. See the input format and the sample input-output for more details.

输入输出格式

输入格式

The first line of the input contains three space-separated integers $ n $ , $ k $ and $ q $ ( $ 2 \le n \le 5000 $ ; $ 1 \le k \le 5000 $ ; $ 1 \le q \le 2 \cdot 10^5 $ ). The second line of the input contains $ n $ integers $ a_1, a_2, \dots, a_n $ ( $ 1 \le a_i \le 10^9 $ ). $ q $ lines follow. Each line contains two space-separated integers $ i $ and $ x $ ( $ 1 \le i \le n $ ; $ 1 \le x \le 10^9 $ ) indicating that you must change the value of $ a_i $ to $ x $ .

输出格式

Print $ q $ integers. The $ i $ -th integer should be the sum of values over all good paths after the first $ i $ updates are performed. Since the answers may be large, print them modulo $ 10^9 + 7 $ .

输入输出样例

输入样例 #1

5 1 5
3 5 1 4 2
1 9
2 4
3 6
4 6
5 2

输出样例 #1

62
58
78
86
86

输入样例 #2

5 2 5
3 5 1 4 2
1 9
2 4
3 6
4 6
5 2

输出样例 #2

157
147
207
227
227

输入样例 #3

4 40 6
92 21 82 46
3 56
1 72
4 28
1 97
2 49
2 88

输出样例 #3

239185261
666314041
50729936
516818968
766409450
756910476

说明

In the first example, the good paths are $ (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4) $ . Initially the values of $ a $ are $ [3, 5, 1, 4, 2] $ . After the first update, they become $ [9, 5, 1, 4, 2] $ . After the second update, they become $ [9, 4, 1, 4, 2] $ , and so on.